# Beam Theory Explained - How Spars Work

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#### wsimpso1

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Hmm, well, I could do one example of the air loading, shear, bending moment, curvature, and deflection. I could even do the exact solution and a numerical integration to show how close they are.

To get into geosnooker2000's questions:

• The air load falls off toward the tips - air leaks around the end. Even a rectangular wing tends to look like it has an elliptical distribution of spanwise load. That complicates things a bit. Then if the wing is tapered or has washout either by twisting the wiing or by using a more highly cambered foil towards the tip, things get more complicated;
• The airload for the wing does carry all the way to the centerline, even when a bulky fusealge is in there. The pressures developed on the wing can not just disappear at the wing, so the pressure distribution looks elliptical all the way to the centerline, and we get our airloads from that;
• The airload summed up on the entire span is actually equal to the weight of the airplane times the g's plus whatever download the tail is making to neutral out the airplane pitching moments. If the tail is making positive lift, it does then reduce the lift needed from the wing, but you do not find much of that except in canard ships;
• In a fabric covered wing, the ribs do collect the airloads and drop that load in at the rib mounts. If you have a plywood D-tube, then the skin ends up distributing the some of the lift along the spar, with a reduced point load at each rib mount. The errors from either going smooth with an exact analytical approach or doing the wing piece-wise at the rib spacing are small, as are the errors from adding fidelity or not from how the load is collected and applied to the spar. Get into composites and we tend to either have one continuous rib (massive foam core) or very few ribs, and the lifting load is nicely spread out onto the spar;
• In all skinning approaches, skin strength and airloads end up setting how far apart we can allow ribs to get... A whole 'nother topic.

I will put together some sort of an example, but it may be a few days out.

Billski

#### geosnooker2000

##### Well-Known Member
Thank you sir! I look forward!

#### Dave Hodges

##### Well-Known Member
So, for the dummies in class, I think it would be instructive for someone who has done one before, to run through an example calculation on an actual spar in an actual wing, showing how the loading diagram was arrived at. For my part, I will jump out and show my ignorance by guessing for y'all's amusement how a spar is loaded, just to demonstrate what some of us need help in understanding.

Keep in mind, I have a masters in Architecture, which means I had 3 semesters of structures, so I am coming at this with calculating beams mostly intended to withstand only forces in the Z axis (with the exception of steel columns - something about radius of gyration that I forgot as soon as I passed the class). We DID get into truss design, so I "had" a good grasp of vector addition, but boy do I need a refresher...

Example. a low-wing aluminum airplane with a span of 32'. Fuselage is 4' wide. Leaves us with a 14' cantilevered beam (Wing Spar). Assuming a tapered wing design, the loads in the positive G category would be a gradual increase as they reach the attached end. There would not be a point load on the outboard end of the beam diagram, even though I imagine picking up a model airplane by the tips of the wings to see if it is within acceptable CG - That is not how the air (wind) will pick up the plane. it will pick it up at every instance along the underside of the spar.

So my first thought was, yeah, it's just a unequal uniform load consisting of half the weight of the plane times however many G-forces you are designing for (times 1.5 safety margin). Then I thought, no... the wing ribs will intersect and attach at, for example, every 16" o.c. Maybe it should be a series of point loads, each point load conveying the area each rib caries. In a building, I would not make the distinction, as wood and steel are far to stiff to worry about buckling @ 16" o.c. braced, but we are talking about aluminum here.

Well, that should be enough to start a beat-down-fest of my ignorance. Have I got ANY of that right?
If you want to brush up on beam design, I hit on it in one of my calculus videos. I forget which one. But if you want to get started, here you go:

#### wsimpso1

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Ok, I ran through an example with elliptical lift distribution, tailored spars, and numerical integration, then thought maybe I ought to give a simpler example first. I still ran a beam supported in the middle but with the load uniformly distributed over the span, figured out shear and bending moment, designed a straight beam section to support all that load, and then figured angles and vertical deflection.

Now if you still want to see all that other stuff, I can proof read it and make it decent and set that up on here too, but I will not bother if you do not ask for it. Grin.

PS - I caught a couple errors, and they are now fixed. If you have already loaded a copy of the attached files, please discard them and copy a fresh file.

Billski

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#### geosnooker2000

##### Well-Known Member
I love the way you can use Excel for this kind of stuff. Maybe my lack of sleep, but I opened up both sets of docs, and they seem to be the same figures.

#### wsimpso1

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I opened up both sets of docs, and they seem to be the same figures.
Fixed it.

#### BBerson

##### Light Plane Philosopher
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I don't use Excel. Using my hand calculations and a sample procedure on page 411 of Aircraft Structures, I find that .020 2024 web is enough for 500 pound gross ultralight. Seems too thin, but 500 is a quarter of 2000 pounds and .020 is a quarter of .080.

#### wsimpso1

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I don't use Excel. Using my hand calculations and a sample procedure on page 411 of Aircraft Structures, I find that .020 2024 web is enough for 500 pound gross ultralight. Seems too thin, but 500 is a quarter of 2000 pounds and .020 is a quarter of .080.
"Other researchers have come to similar conclusions" translates to "somebody else thinks so too". hehehe...

We still have to prevent buckling, which in both of these webs will likely be an issue too. There are probably stiffeners in the future for both of these spars unless there are a lot of ribs...

Billski

#### proppastie

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This is a fair amount more complicated.

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#### BBerson

##### Light Plane Philosopher
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My example calculation was with stiffeners at 5" spacing. Which gave a factor or safety of 2.4. Don't know if that is non- buckling or limited elastic buckling.
Thanks for the Bruhn chapter.

#### geosnooker2000

##### Well-Known Member
Check out the thickness on these main spars at the business end. What do you suppose that thickness is? In total, and individual sheets. Are those just gussets that penetrate into the first web? And 4 bolts at the bottoms and 4 bolts at the tops? That's all you need to stave off roughly 6000 lbs of shear? That's 6000 lbs / 8 bolts = 750 lbs per bolt... I guess that seems reasonable. I would have just thought those connections would go all the way to the mid-point of the fuselage (2' long) under the seats and connect with a gusset that is the whole width of the cabin (4').

"quick build" wings for a Sling 4

#### wsimpso1

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Something to know about wing spars. At the root, shear is the accumulated lift of that wing, while bending is the lift times the distance to about 40% out on the wing. Moments are huge compared to shear... That width you see at the top and bottom are probably the width of the caps need to carry the bending load converted into tensile load (in one cap) and compressive load (in the other cap). The lateral loads through the bolts will be an order of magnitude larger than the vertical loads. The shear webs are probably that thick at the joints simply to help collect the loads to the bolted joints.

Metal wings are frequently done this way. We can collect the loads and carry them compactly - that is in small volume - in metal structures. In composite wings hardpoints tend to be really bulky, so we tend to overlap spars to get the forces down and then use only a couple bolts or large diameter thin wall pins.

There are what I count as four bolts top and four more bottom. The bolts are in shear twice each, once on each side of the spar, so there is 16 times the area of one bolt. It is common to design the joints with bolts or rivets to a FOS of 4 while the rest of the aluminum structure uses FOS of 1.5. And they appear to be AN6 bolts, which are 3/8 diameter and 125,000 psi steel. That works out to about 8000 pounds of shear strength per shear surface... The aluminum is probably a 33,000 or 37,000 psi material, and so it needs more bearing area than the bolts do... Shigley is the book to go to for an intro to bolted joint design.

The Sling aircraft work fine. You want to get freaked out, look at a C172 in the maintenance shop with the root covers off. One bolt at the main spar, one at the drag spar, and two little AN bolts at each end of the lift strut. That's all it needs and they work fine... But each of these designs was engineered carefully.

Billski

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#### wsimpso1

##### Super Moderator
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I just put the Sling 4's specifics into the spreadsheet I supplied, added in a calc for bolt size needed and that stuff all looks about right...

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#### geosnooker2000

##### Well-Known Member
Note that, when I said 6000 lbs of shear, I got that from total w @ a cantilever beam support end= shear
1000 lbs (half the max gross weight) x 4Gs = 4000 lbs x 1.5 factor = 6000 lbs.

#### wsimpso1

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Note that, when I said 6000 lbs of shear, I got that from total w @ a cantilever beam support end= shear
1000 lbs (half the max gross weight) x 4Gs = 4000 lbs x 1.5 factor = 6000 lbs.
I agree with the 6000 pounds of shear.

My point is the big load on the bolted joints is not the shear. It is the bending moment at someplace around 350 - 400,000 in-lb. If the two lines of bolts are 6" apart, the linear force on each line is about 59 -67,000 pounds. The shear at 6000 is at right angles to the bending reaction, so they add by sum of squares (sqrt(a^2 + b^2) = c) or still about 59-67,000 pounds...

Billski

#### geosnooker2000

##### Well-Known Member
So, here's something to think about. If the inboard end of the spar transitioned to a hollow tube (not pipe {round}, but rectangular tube), and the 8 bolts were moved 90 degrees to bolt the top and bottom flanges in a vertical fashion, would the shear forces be any different? The top joint would be in compression, the bottom joint in tension, but the result would still be two pieces of metal trying to slide past each other putting a shearing force on the bolts.

#### wsimpso1

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So, here's something to think about. If the inboard end of the spar transitioned to a hollow tube (not pipe {round}, but rectangular tube), and the 8 bolts were moved 90 degrees to bolt the top and bottom flanges in a vertical fashion, would the shear forces be any different? The top joint would be in compression, the bottom joint in tension, but the result would still be two pieces of metal trying to slide past each other putting a shearing force on the bolts.
You are playing "what if" on fitting and joint design on a thread about how beams work. This is thread drift.

If you want to discuss fitting and joint design, please start a thread on the topic and ask there. In the meanwhile, arm yourself with some knowledge - see if you can find a book on Strength of Materials and/or Shigley's book on Mechanical Engineering Design. Libraries can get them on interlibrary loan or you can buy (used) earlier editions pretty inexpensively. With a little study of the chapters on beam theory and joint design, you will be able to propose and answer many "what if" scenarios yourself. Understand that beams and joints and fasteners were figured out long before the Wrights invented airplanes. The possibility of figuring out something new in these topics is remote. You will be well served by being able to figure out beams and joints and fasteners as we understand them.

Billski

#### geosnooker2000

##### Well-Known Member
I will shut up. But understand, I have had three semesters of structures in Architecture school, I know how to calculate and size beams, I have done it for 34 years. I have designed steel buildings, wood frame buildings, concrete buildings (including parking garages, footings, bridges, etc.). But this (wing design/spar design) is far more complicated than earthbound buildings of any sort. Because not only does the wing have a vertical load on it, which is the same as any cantilevered beam I've designed in the past, it also has a horizontal load from drag forces on the plane which is not the same as a building. On a building, those forces (such as seismic and wind loads) are always negated by bracing. The only bracing against drag forces is the skin of the wing (and possibly internal "x" braces which are not used in all wings). Is that enough? Is that what is totally relied upon to negate side load on the spar? I don't know. This is what makes me think this is not as simple as a cantilevered balcony with 2 steel beams holding up a floor joist system between them.

Personally I like conversation. I am self-employed and have very little human contact other than my wife and my clients, so I prefer to converse about things with people whom share my interests (mostly on message boards) instead of study them in books. Not to say that I won't. I just prefer the former.

If you think discussing the end connections of a beam is a "thread drift" in a thread about BEAMS, then I don't know what to tell you...

#### mcrae0104

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Geo, grab a copy of Stress Without Tears. It will pick up where your structural education left off. You are right that wings are not as simple as cantilevered balconies, but the good news is that once you really understand the underlying principles, you are equipped to solve either problem.

#### wsimpso1

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Please pardon my direction to go to the books. We can not hope to teach big chunks of undergraduate semesters on a forum. Topics are just too big to do without text, examples, and then practice.

I do feel that folks who follow this forum should be able to find the topics of discussion by title. The genesis of this thread was a number of folks talking about beams while it was obvious that they did not understand the pieces of beam interact... I am happy to have another thread talking about how to connect things together, and another on how wings carry loads, but they are different topics. Trying to meld beam design and hardpoint design in one thread, well, it complicates learning for those at the beginning of the process. Let's just do one thing per thread...

I am starting a thread on wing and spar interrelationships...

Billski

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